One major reason that Fourier transforms are so important in image processing is the
convolution theorem which states that

If f(x) and g(x) are two functions with Fourier transforms
F(u) and
G(u), then the Fourier transform of the convolution f(x)*g(x) is simply the
product of the Fourier transforms of the two functions, F(u) G(u).

Thus in principle we can undo a convolution. e.g. to compensate
for a less than ideal image capture system:

Take the Fourier transform of the
imperfect image,

Take the Fourier transform of the function describing the effect of the
system,

Divide the former by the latter to obtain the Fourier transform of
the ideal image.